Sunday, January 25, 2015

Using Algebra to Derive New Formulas and Solve others

To follow this you are going to need to know some algebra.
If you need a refresher, I recommend www.mathisfun.com. I am first going to show how to
use algebra to find the temperature at which both a Fahrenheit and Celsius
thermometer would be showing the same number. That is, what temperature
is the same in degrees Fahrenheit and Celsius. You can think of this like
using 2 different rulers to measure your height. One in feet and one in
meters. Your height is the same no matter which you use, just the number
will be different, i.e. 6 feet or 182.88 centimeters.

First we start with the conversion formulas. To convert from
degrees Celsius (C) to Fahrenheit (F), we multiply the temperature by 1.8 (also
regularly stated as 9/5) then add 32. So 20 degrees C = 68 degrees F.

Converting from F to C the formula is degrees F minus 32 then that
result divided by 1.8:

To
calculate the temperature where these are equal to each other, we set the 2
formulas equal to each other, use just one variable for temperature, e.g. “t” and solve for that variable:

The
idea is to get the t all by itself
and we do this by performing the inverse operations on various parts of the
equation, remembering to do the same operation on BOTH sides! First we’ll get rid of the 1.8 on the bottom
of the right side by multiplying both sides by 1.8:

The
1.8’s on the right side cancel leaving just the t - 32 and we multiply the 2 terms inside the
parenthesis on the left side each by 1.8, giving us:

Now
subtract t from both sides:

Doing
the math this leaves us with:

Next
we subtract 57.6 from both sides:

This
gives us:

Next
we divide both sides by 2.24 to get the t
by itself (this is the same as multiplying by 1/2.24):

The
result of this is:

You
can check this by inserting minus 40 into the 2 equations above and you’ll see
the conversions both come out to minus 40.

Now
as they say, I showed you that so I can show you this, specifically, how to
derive a new formula.

In
the license books we’ve seen examples, usually poorly explained, where a new
formula is derived by substituting values.
In the case of the power formula, P = E I, we are told to substitute the
ohms law equivalents of E and I to get 2 new formulas.

The
example I’m going to show you is from the ARRL Extra Class license manual which
is only shown worked through superficially.
The assumption being that either the student is well versed in algebra,
or will go to one of the many resources available online or elsewhere to bone
up on it.

However,
I have found that many who come to ham radio, especially those who come to it
after a prolonged time away from high school or college, may have forgotten
most, if not all that they’ve learned.
So I’d like to build on the previous example and work through the
derivation of the formula for calculating resonant frequency of LC circuits.

The
definition of LC circuit resonance is given as the frequency where both inductive
reactance (XL) and capacitive reactance (XC) are
equal. Each reactance being 180 degrees
out of phase with the other means that their reactances will cancel out leaving
only a purely resistive load. Let’s
first look at the formulas for each, then we’ll set them equal to each other
and solve for frequency to get the resonant frequency formula.

Setting
them equal:

We
want to find “f” so we’ll multiply
both sides by f to remove it from the
bottom of the right side.

Combining
the f’s on the left and cancelling on
the right leaves us with:

Next
I’m going to get rid of the 2πL on
the left by dividing both sides by 2πL
(this is the same as multiplying by the reciprocal 1/2πL)

When
we do the math we cancel the 2πL on
the left side and multiply the top and bottom on the right, combining the like
terms to get: